Uses of power series as opposed to taylor series

[gsingh2011]

So we can use the Taylor's theorem to come up with a Taylor series represent certain functions. This series is a power series. So far (I'm in my second year of calc, senior in high school), I've never seen a power series that wasn't a Taylor series. So are all power series taylor series? Whether the answer to that is yes or no, what are power series used for independent of taylor series?

 

[zetafunction]

appart from Taylor series there are ORTHOGONAL SERIES of polyonomials i mean

$$\int_{a}^{b}dxT_{n}(x)T_{m}(x)u(x) = \delta _{n,m}$$

so you could expand any function f(x) as

$$f(x)= \sum_{n=0}^{\infty}c_{n} T_{n} (x)$$ (1)

here T(x) are POlynomials so (1) can be regarded also as a power series different from the Taylor one

 

[HallsofIvy]

A power series is a power series is a power series!

A "Taylor series" is a particular way of getting the power series representing a particular function.

For example, if you are asked to write $(1- x)^{-1}$ as a power series in x, there are two ways to do that:

1) Find the Taylor's series for 1/(1- x) around x= 0 (the MacLaurin series) by taking the derivatives.

2) Recall that the sum of the geometric series $\sum a r^n$ is given by a/(1- r) so that $1/(1- x)= \sum x^n$.

Those are two different ways of forming a power series but they give exactly the same power series for the same function. Even the series of polynomials that zetafunction refers to, once you combine like powers, are the same power series.